Maxwell's equations, nonlinear media, and dynamic response

Question

Maxwell's equations in the vacuum with electric permittivity $\epsilon_0$ and magnetic permeability $\mu_0$ are given as:

$$\nabla \cdot \vec E = \frac{\rho}{ \epsilon_0}$$ $$\nabla \cdot \vec B = 0$$ $$\nabla \times \vec E = - \frac {\partial \vec B}{\partial t}$$ $$\nabla \times \vec B = \mu_0 \vec J + \epsilon_0 \mu_0 \frac {\partial \vec E}{\partial t}$$

In material media, $\epsilon$ and $\mu$ are larger or smaller than $\epsilon_0$ and $\mu_0$ and may depend on $\vec x$ and even on the direction of polarization.

All that seems okay to me at first glance. However, in nonlinear media, $\epsilon$ and $\mu$ depend on $\vec E$ and $\vec B$. So, for nonlinear media, Maxwell's equations are often written as:

$$\nabla \cdot \vec E = \frac{\rho}{ \epsilon (\vec E,\vec B)}$$ $$\nabla \cdot \vec B = 0$$ $$\nabla \times \vec E = - \frac {\partial \vec B}{\partial t}$$ $$\nabla \times \vec B = \mu (\vec E,\vec B) \vec J + \epsilon (\vec E,\vec B) \mu (\vec E,\vec B) \frac {\partial \vec E}{\partial t}$$

(As a further generalization, $\epsilon$ and $\mu$ are sometimes represented as tensors whose components are functions of $\vec E$ and $\vec B$, but that is not an important issue for the current question.)

My problem with the nonlinear material media version of Maxwell's equations is that it seems to assume instantaneous material response to changing $\vec E$ and $\vec B$, while it seems that any physically plausible material can only respond in finite time. It would be like saying that a spring's length is proportional to the force applied- which is true only when the applied force is changed very slowly. That is, I expect that any real material will have a dynamic response to changing $\vec E$ and $\vec B$.

If that's true, then it seems it should make better sense for the specification of $\epsilon$ and $\mu$ to be in the form of differential or integral equations including time. Of course that would complicate the math a lot, but from a physics perspective it would be more plausible. My question: Is there a form of Maxwell's equations in nonlinear media that takes the dynamic response of the medium into account? A follow-up question would be "Is there a Lorentz-covariant form of those equations?"

Answer 1

What you are talking about is dispersion. Dispersion is not necessarily nonlinear phenomenon, it occurs in linear media as well. Moreover you can have spatial and temporal dispersion. Temporal dispersion means that system response depends on what is the stimulus is currently as well as on what it was earlier. Spatial dispersion means that you material response at position A depends on what the field does at position $B\neq A$

There are many ways to account for these phenomena, I will only list how it is done in dielectrics. Other generalizations are similar

In non-trivial dielectics you would have

$\boldsymbol{\nabla}.\mathbf{D}=0$

$\boldsymbol{\nabla}.\mathbf{B}=0$

$\boldsymbol{\nabla}\times\mathbf{E}=-\partial_t\mathbf{B}$

$\boldsymbol{\nabla}\times\mathbf{B}=\mu_0\partial_t\mathbf{D}$

Now you can stick all you complex material response into displacement. Want temporal dispersion (linear case)? Here you go:

$\mathbf{D}\left(t,\mathbf{r}\right)=\epsilon_0\int^t_{-\infty}dt' \boldsymbol{\epsilon_r}\left(t-t',\mathbf{r}\right).\mathbf{E}\left(t',\mathbf{r}\right)$

Spatial dispersion (linear)?

$\mathbf{D}\left(t,\mathbf{r}\right)=\epsilon_0\int d^3r' \,\boldsymbol{\epsilon_r}\left(t,\mathbf{r}-\mathbf{r'}\right).\mathbf{E}\left(t,\mathbf{r'}\right)$

For non-linear response you play similar games but you tend to use polarization density, i.e. $\mathbf{P}=\mathbf{D}-\epsilon_0\mathbf{E}$. Second order non-linearity with temporal dispersion:

$\mathbf{P}\left(t,\mathbf{r}\right)=\int^t_{-\infty} dt'\int^t_{-\infty} dt'' \boldsymbol{\chi^{(2)}\left(t-t',t-t'',\mathbf{r}\right)}.\mathbf{E}\left(t',\mathbf{r}\right).\mathbf{E}\left(t'',\mathbf{r}\right)$

etc... Most books on nonlinear optics will cover this

PS: $\epsilon_r$ is the relative perimittivity tensor, $\chi^{(2)}$ is the second-order susceptibility tensor.

Answer 2

Is there a form of Maxwell's equations in nonlinear media that takes the dynamic response of the medium into account?

Yes there is, but it probably won't satisfy you. The general form is the same as the usual Maxwell's equations for fields in the presence of known charge and current distribution in vacuum. The only thing the material medium is assumed to change is that distributions $\rho,\mathbf j$ have another contribution due to material medium.

Such Maxwell's equations are not a complete system of differential equations, it is instead an underspecified system, so some other assumptions must be introduced and used to relate distributions of charge and current on one side, and EM fields on the other side.

These assumptions vary with the physical situation such as static dielectric polarization ($\mathbf P$ is sufficient), static ferromagnet magnetization ($\mathbf M$ and $\mathbf j_{free}$ is sufficient), or high frequency dissipative EM wave propagation (one better works with some microscopic model and $\rho,\mathbf j$ directly). They also depend on the quality of the material medium, which there are a lot of kinds. There is no general formulation of EM theory of material medium that would provide a closed system of equations.